Question: Simplify; express your answer in exponential form. Assume $t\neq 0, r\neq 0$. $\dfrac{{(t^{-3}r^{3})^{-4}}}{{(t^{-1}r^{-5})^{-4}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{-3}r^{3})^{-4} = (t^{-3})^{-4}(r^{3})^{-4}}$ On the left, we have ${t^{-3}}$ to the exponent ${-4}$ . Now ${-3 \times -4 = 12}$ , so ${(t^{-3})^{-4} = t^{12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{-3}r^{3})^{-4}}}{{(t^{-1}r^{-5})^{-4}}} = \dfrac{{t^{12}r^{-12}}}{{t^{4}r^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{12}r^{-12}}}{{t^{4}r^{20}}} = \dfrac{{t^{12}}}{{t^{4}}} \cdot \dfrac{{r^{-12}}}{{r^{20}}} = t^{{12} - {4}} \cdot r^{{-12} - {20}} = t^{8}r^{-32}$